...An unmodified, traditional longbow usually hits it's maximum at about 350m, while the footbow record is very near 1850m (over a mile).
These numbers seem suspect to me.
I can certainly believe that 350m is about the limit of the useful range of a longbow when it's considered as one man aiming at a single standard target. However, I have some difficulty believing that someone lying down on the ground to use a footbow would be able to reliably hit a specific man-sized target at a range of 1850m, which is the criterion that should be used if we are comparing like with like.
According to
World Record archer, Barry Groves, article on Flight Archery - the Pursuit of Perfection, the distance record with a conventional (ie not compound, although it does not specify if composite recurve or self) is 1222m.
http://hyperphysics.phy-astr.gsu.edu/ has some nice formulae in the trajectory section that are relevant. Assuming an initial arrow speed of 75 m/s (246 fps, slightly below that of modern construction self bows), we get the following theoretical ranges:
* 1020 m at 45° angle (optimal for range)
* 884 m at 30° angle
Doing a reverse-calculation, to have a maximum theoretical range of 350m, the launch velocity would have to be no greater than 58.57 m/s (192 fps), which is significantly below the velocity of even the slowest arrows that have actually been measured.
Backcountry Bowhunting - Archery Calculator has a nice calculator for determining expected arrow speed.